APPLICATIONS OF LIE THEORY. Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543. Email [email protected] Tel (65) 6516-2749. http://www.math.nus.edu.sg/~matwml/courses/Undergraduate/USC/2006/USC3002. THE UTILITY OF LIE THEORY.
Lie theory models nonlinear constraints and symmetry
that explains phenomena and arises in engineering
design and simulation of physical processes.
Shlomo Sternberg - Group Theory and Physics
1928 Weyl’s Gruppentheorie und Quantenmechanik
1920-30’s Chemistry and Spectroscopy
1930-40’s Nuclear and Particle Physics
1960-70’s High Energy Physics
Sternberg notes: the remarks on pages 60-2 in Slater, J.C.Solid-State and
Molecular Theory: A Scientific Biography. New York: Wiley, 1975.
“It was at this point that Wigner, Hund, Heitler, and Weyl
entered the picture with their “Gruppenpest”: the pest of group
theory…The authors of “Gruppenpest” wrote papers that were
incomprehensible to those like me who had not studied group
theory, in which they applied these theoretical results to the
study of the many electron problem. The practical
consequences appeared to be negligible, but everyone
felt that to be in the mainstream one had to learn about it.”
and remarks “It is, however, amazing to consider that this
autobiography was published in 1975, after the major
triumphs of group theory in elementary particle physics.”
of the utility of Lie theory in engineering design and physical
simulation may be slower than it was in physics, due to
1. the extensive mathematical background of physicists, and
Slater, John Clarke (1900-1976) American physicist who did work on the application of quantum mechanics to the chemical bond and the structure of substances. His name is attached to the Slater determinant used to construct antisymmetric wavefunctions.
2. the fundamental nature of physics. But the success of Lie
theory in physics may facilitate recognition in other fields.
Calderbank and Moran have demonstrated the importance
of discrete Heisenberg-Weyl groups in coding.
Lie symmetry explains soliton/instanton phenomena that have
growing importance in material science and nanotechnology.
were chosen based on 1. my interests and 2. instructive value.
Functions with values in Lie groups
Functions (measures) on Lie groups
(for waves in random media)
whose Fourier transform
or frequency response
satisfies the nonlinear constraint
Definition F has regularity N if
Regular CQF’s used for filterbanks & wavelets
and there exists a
whose 1st column equals
is a trigonometric polynomial (finite
impulse response filter) iff every component of
can be chosen to have trig. poly. entries.
(in Pressley and Segal’s Loop Groups)
can be uniformly
approximated by a trig. poly.
Proof. Trotter formula and semisimplicity of SU(m)
Every continuous CQF
can be uniformly approximated by a trig. poly. CQF
with the same regularity as
Proof. , uses Lemma 1, Jets, and the Brower degree
Flow is a trajectory
in the group of volume-preserving diffeomorphisms.
is the angular velocity in space
Moreau 1959 g is a geodesic with respect to the
right-invariant kinetic-energy Riemannian metric
For d = 2, div u = 0 a Stream function
SDiff(D) is the Lie algebra with Poisson bracket
Flow preserves Casimirs
For periodic flow basis
gives algebra derivations
algebra (non comm. torus)
generated by A,B where
,N odd gives homomorphism onto
with real form
Sine-Euler Approximation by GF on
N-Casimirs are exactly conserved ! See 
in a chain of harmonic oscillators
is described by the equation
admits an expansion in normal modes
are samples of a measure on
the Lie group
Theorem Localization of
Proof Furstenberg  the Lyapunov exponent
 W. Lawton, Hermite interpolation in loop groups
and conjugate quadrature filter approximation, Acta
Applicandae Mathematicae,84(3),315--349, Dec. 2004
 V. Arnold and B. Khesin, Topological Methods
in Hydrodynamics, Springer, New York, 1998.
 H. Furstenberg, Noncommuting random products,
Transactions of the AMS, 108, 377-429, 1963.